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SCALAR POTENTIAL AND DYONIC STRINGS

IN 6D GAUGED SUPERGRAVITI r

S. Randjbar-Daemi

and

E. Sezgin

Available at: http: //www. ictp. trieste. it/-pub- of f IC/2004/6 MIFP-03-25

United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SCALAR POTENTIAL AND DI.rONIC STRINGS IN 6D GAUGED SUPERI-TRAVITY

S. Randjbar-Daemi The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

E. Sezgin George P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843-4242, USA.

Abstract

.In this paper we first give a simple parametrization of the scalar coset manifold of the only known anomaly free chiraJ gauged supergravity in six dimensions in the absence of linear multiplets,

namely gauged minimal supergravity coupled to a tensor multiplet, E6 x E7 X U(1)R Yang-Mills multiplets and suitable number of hypermultiplets. We then construct the potential for the scalars and show that it has a unique minimum at the origin. We also construct a new BPS dyonic string solution in which U(1)R X U(1) gauge fields, in addition to the metric, dilaton and the 2-form potential, assume nontrivial configurations in any U(1)R gauged 6D minimaJ supergravity coupled to a tensor multiplet with gauge symmetry G Q U(1). The solution Preserves 14 of the 6D supersymmetries and can be tivially embedded in the anomaly free model, in which case the U(1) activated in our solution resides in E7.

MIRAMARE - TRIESTE February 2004 1 Introduction

The most symmetric ground state solutions in all of the higher dimensional supergravity theories, as the low energy limit of the superstring theories or the M theory, are the flat 10 dimensional manifolds Rd x f lat torus) and the pp waves. Such backgrounds have 32 real supersymmetries and are not a suitable starting point for building realistic phenomenology. The Calabi Yau compactifications have less supersymmetries but also a lesser degree of uniqueness. There exist many of them with a lot of moduli with unknown potentials.

In the gauged minimal supergravity theories in D = 6 this is not the case. In fact in these theories the 6-dimensional flat spaces do not solve the supergravity equations. The most symmetric solution is R4 X S2 [1] which has been shown recently to be the unique maximally symmetric solution of such models 2]. This result has been obtained essentially in the minimal version of such models for which the gauge group is simply U(1), i.e. the D = 6 supersymmetric Einstein Maxwell theory, known as the Salam-Sezgin model in the literature [1]. The model by itself is anomalous but it can be embedded into an anomaly-free model 3] with suitable Yang-Mills and hypermatter sector couplings 4, 5].

The uniqueness of the supersymmetric R4 X S2 solution should be contrasted with the plethora of solutions of Calabi Yau type or the ones with exceptional holonomy groups such as G in higher dimensional low energy string theories. For this reason we consider this property as very interesting and believe that such models deserve further study. This is the aim of the present note.

This paper contains two results. First, after giving the multiplet structure of general gauged N = 1 supergravity models in D = 6 we shall concentrate on the hypermatter sector. This sector contains the scalars and fermions which can be in anomaly free representation of the Yang-Mills gauge groups and therefore axe of primary importance in any phenomenological application of such models. So fax we know of only one anomaly free gauged minimal supergravity in D = 6 3] in the absence of linear multiplets 1. For this reason we shall construct in detail the scalar manifold and the potential for the scalars in this particular model. However, our simple parametrization of the quaternionic scalar manifold and the construction of the scalar potential should be applicable in other cases too. The first result of this construction is the observation that the scalar potential admits a unique minimum at the origin. There are no moduli. As we shall see this trivially implies the uniqueness of the R 4 X S2 solution in the anomaly free full fledged model.

The second main point of the paper is the construction of new dyonic string solutions. These

'The linear multiplet is a hypermultipIet in which one of the four scalars is dualized to a 4-form potential. In this case, an additional Green-Schwarz counterterm is possible for anomaly cancellation 6], and this may lead to new anomaly free models.

2 :solutions have a rather nontrivial structure and leave 14 of the D = 6 supersymmetries unbroken. The search for such solutions is motivated by the general philosophy that they can help us -to study string and field theories from a non-perturbative, semiclassical point of view.

'The choice of the model to be considered here is dictated by anomaly cancellation. It belongs to -a general class of (1,O) supergravity models constructed some time ago 4, 5]. It is based on a six- ,dimensional (1, 0) supergravity theory coupled to a tensor, Yang-Mills and hyper multiplets 3]. At present this is the only known gauged (1, 0) anomaly free supergravity in D = 6 in the -absence of linear multiplets. Its string or M-theory origin is still not quite well understood, although some progress has been made in connecting a subset of the model to M-theory in 11- -dimensions 7], and Horava-Witten type construction in 7-dimensions [8]. Note, however, that in the latter case the R-symmetry group is not gauged i the resulting D = 6 supergravity.

Recently, our model has attracted interest in connection with a possibility of obtaining a small cosmological constant in D = 4 9]. These ideas make use of an extension of the old magnetic monopole solution [10] to a situation in which 3-branes are distributed over a transverse S2 [11, 12]. In order for the mechanism to work it is required that the supersymmetry breaking on the brane does not propagate to the bulk. This is the weak point of the scheme for a finite volume 2-manifold, as has been argued in a very general context in 13]. We hope that some other variations of the idea will make a dent on this very important unsolved problem.

All the supersymmetric solutions of our model known so far, whether compactifying solutions with a direct product geometry or more involved stringy solutions, use only the UMR gauge fields as part of their ansatz. It seems quite difficult to excite the gauge fields in the non-Abelian E6 x E7 component of the gauge group in search of a supersynnnetric solution. In this paper we report on one such solution in which the gauge field in a U(1) subgroup of E7, in addition to that of the U(1)R factor in the gauge group, is also nonzero. The configuration is quite involved and does not have a 4-dimensional Poincar6 invariance. It has a natural dyonic stringy interpretation similar to the one in [14]. In fact it reduees to the solution in [141 upon setting the E7 gauge field to zero.

'Thus, in our dyonic string solution, in addition to the teELsor fields and the gauge field of U(1)R, a U(1) gauge field embedded in E7 as well as the dilaton field are also active. The only fields which do not participate in the solution are the E6 gauge fields and the hyperscalaxs.

The composition of the paper is as follows: In section 2 we give a detailed description of the model with explicit form for the potential in the hyperscalars. In this section we also give the field equations and the supersymmetry transformation rules. In section 3 we show that the -absolute minimum of the scalar potential is at 0 = 0. This fact excludes a solution of the form M4 x K2 with a nonzero vev for any of the gauge fields in E6 x E7 and with any unbroken susy. [n section 3 we discuss the ansatz for the dyonic string with a nonzero vev for the gauge fields

3 in U(1)R x E7 as well as the tensor fields. In section 4 we verify that our ansatz satisfies the field equations and leaves 14 of the original supersymmetries, i.e. one complex susy in 1 + 1

dimensions, unbroken. In the limit of a vanishing E7 field it reduces to the solution found earlier in 14] where only U(1)R field was activated. In this section we also show that our solution has

a horizon at = while at r = oo it approaches a cone over a squashed S3 X Minkowski2- The dilaton diverges in both limits. Section contains our conclusions. We give the anomaly polynomial in an Appendix correcting the misprints of 3].

2 The Model

2.1 Field Content and the Scalar Manifold

The six-dimensional gauged supergravity model we shall study involves the following N (1, 0)

supermultiplets 2 graviton er oA B- M MN tensor(dilaton) W XA BMN hypermatter V)' YangMills Am AA+ where coordinate basis and tangent space indices are denoted by M, N,... and r, s,..., respectively. The antisymmetric tensor potentials, B4N1 give rise to selfdual and anti-selfdual 3-form field strengths. All the spinors are symplectic Majorana-Weyl A = 2 label the doublet of the R symmetry group S(1)R and a = 912 labels the 912 dimensional pseudoreal representation of E7. The chiralities of the fermions are denoted by ±.

The hyperscalars O' a 912 x 2 parameterize the quaternionic Kahler coset

Sp(456, 1) (2.2) Sp(456) X S(1)R

The global Sp(456, 1) symmetry is broken by the gauging of its compact E7 x U(1) subgroup.

The composite local Sp(456) X SAOR symmetry is left intact by this gauging. Thus, together with the "external" E6 gauge symmetry, the full symmetry of the model is

[E7 E6 U(1)11.cal x [Sp(456) S)Icomposite local (2-3)

Note that E7 in Sp(456) is such that the 912 of Sp(456) is irreducible under E7. This spectrum is anomaly free 3 and so fr is the only known anomaly free gauged supergravity model in D = 6 in the absence of lineax multiplets.

'There also exists a linear multiplet consisting of a 4-form potential, a sympIectic Majorana-Weyl spinor and three real scalars but it is not coupled in the model we study here.

4 It will prove useful to present the model of 3] in alternative forms. To this end, we need to introduce some notation and outline the building blocks. To begin with, we define the vielbein, Sp(n) and Sp(l) composite connections on the coset via the Maurer-Cartan form as

)aA = VaA I ab ab I AB AB (L-1aaL a (L- 9L) = Aa (L - 9 L) Aa (2.4)

The vielbeins obey the following relations

9ck,6Va`AbB = 2ab'EAB , V,aA V,6aB + '6 = gafl jAB (2.5)

Where ge is the metric on the coset and S2 and are te symplectic invariant antisymmetric

matrices. Next, we define the components of the E7 X MR gauged Maurer-Cartan form as

(L-'DL) aA . paA (L-'DL)ab = Qab (L-1 DL) AB = QAB' (2.6)

-where

DL= (d-gA 3T 3 - 7A"T' ) L . (2.7)

.Here A3 and A, M (I = 1, ... , 133) axe the MR x E7 gauge fields and g, and 97 axe the corre- sponding gauge coupling constants. The following relations hold

paA = E)oc,)VaA Qal = (,Doc,)A ab QAB:= ('Doc')AAB _ g, A 3(T 3)AB (2.8)

The covaxiant derivative can be written as

Dmoa = i9moa - gjA M3 K 3a - 97A,M KIa (2.9)

where KU' and KI' are the Killing vectors associated with E7 X UMR C Sp(456, 1) isometry.

Other building blocks to define the model are certain C-functions on the coset 2-2). These were

defined in [5], and studied further in [15] where it was shown that they can be expressed as

3 3L CAB = (L-1T ) AB CIA, -'TL)A.11

CaA = 3 aA qaA = -1TIL aA 3 (L-1T L) (L (2.10) where T3 and T, axe the anti-hermitian generators Of U(1)R and E7.

11.2 The Choice of L

From. the foregoing description it is clear that the main ingredient in this construction is the section L, which maps the coset GIH to the group manifold G. We begin thus with a brief description of the groups involved here. Firstly, by Sp(n, 1) what is really meant is the group

(if pseudounitary (2n+2)-dimensional matrices

Sp(n + 1) n SU(2n,'2) (2.11)

5 It is convenient to represent these matrices by (n + l)-dimensional arrays whose elements are 2-dimensional matrices, g (gm') M, v 0, .... n (9,.A vB) A, B 1,2 (2.12)

With this notation we have two kinds of metric:

J = diag 0'2 .... 02 Or2)

7 = diag(12, 12, 12) (2.13) where U2 and 12 denote the matrices

U2 0 i 12 1 0 i 0 0 1

The group elements are required to satisfy two conditions

gtJg = J (symplectic) gtng = 7 (pseudounitary) (2.14) where g and gt denote transpose and hermitian conjugate-in the (2n+2)-dimensional sense- respectively. It is straightforward now to show that each 2 x 2 element in the (n+l)-dimensional array satisfies the reality condition,

gU = U29p Vtt U2 (2.15)

It can be interpreted as a real quaternion.

Having defined the group USp(n, 1) we may restrict to its maximal compact subgroup, USp(n) x USp(l) by choosing go, = gl, 0 = 0 I n

The matrices goo belong to SU(2), i.e.

usp(i) = sp(l) nSU(2) = SU(2)

To coordinatize the manifold, consider the 'boost',

L a+ boot (2.16) Ot c where is a 2n x 2 matrix, Ot is its hermitian conjugate, a, b are real and proportional to the 2n x 2n identity matrix, and is real and proportional to the 2 x 2 identity matrix. We can write

6 'Where the elements of this column are 2 x 2 matrices satisfying the reality condition given above (and repeated below). It follows that Lo belongs to te group USp(n, 1) provided it is also unitary,

Lt 77LO = 77

This is achieved by choosing a b (-I + Vl +O OI/Oto (2.17) C

In obtaining this result we have used the fact that the 2 x 2 matrix Oto is proportional to the identity 12 -

In our problem n = 456 and thus the scalars have 912 x 2 complex components. As real quaternions they can be considered as a 456 x 1 matrixvvhose elements, Om, axe 2 x 2 matrices subject to the reality condition

Om = C2 0. 62, m = 1, 456 (2.18) where o2 is the Pauli matrix, then we have a total of 456 x 4 = 1824 real components. Thus, the following two notations are equivalent:

OaA ++ (Om) Al (2.19) where a = , ... , 912 and A, A' = 1, 2. Using the particular form of the coset representative described above, the following C-functions take a particularly simple form

c3 = 1 112) 3 AB (T )AB CIB = (OITIO)AB , (2.20) where 1012 = troto. These axe the only components we need in order to construct the scalar potential.

2.3 Field Equations and Supersymmetry 1ransformation Rules

The bosonic sector of the Lagrangian is given by [4]

,C = R*1 - 1*dW A dW - le' * HA H - je2Ptr(*FAF) 4 2

1 1,P (tr C2) *1 *Doc' A DOO g,,,6 - 4 e- 2 (2.21) where

H dB+ltrFAA,2 dH=ltrFAF2

tr (*FAF) *F 3 A F 3 + *F, A F, + *F,'A F" I 133, = 1, 2,...,78

C2 2C3 C3,AB + g2 I CIAB tr 91 AB 7CAB (2.22)

7 The bosonic field equation following from the above Lagrangian are 4]

1 2 2 RMN = 4119MW 9NW + 21 e2l tr (FMN - 8 F MN) + 4 el (HM2N 6'H MN)

1 paA C2) 2 MPNaA+e-21(tr 9MN

1 2 2 1 C2 -4e-2(Ptr 41e2ltrF 61elH

1 3) 3_gl*paA 3 d(e2v * F = ew*HAF CaA

1 PaA d(e 2 W* F) = ew*HAF, 97 CaIA

I d(e 2 V * F) = el *HA FI'

d el * H) = 0 ,

,DMpMaA = 2ew 2CAB(C3)a g2CAB(CI)a (2.23) (91 B 7 B)

The local supersymmetry transformations of the fermions axe given by 4]

1 N Jkm = Dme + 481 e 2,PHNpQr PQrm e

JX = 1 (FmOmo - 1 e 12vH- prMNP) 4 6 M

3 = 1 InMN 6A FMN EA - ie 2'OCAB CB

rMN JAIA = - 9FM1N EA - 7e-2'OCAIB CB

JAI'A = - 9dINr MN EA

61Pa = PkArMEA

(2.24)

where DMEA aMEA + 4WM,,]P"-A + QMA B CB-

2.4 The Potential and its Minimum

It is convenient to re-write the hyperscalar field equation as

av (2.25) 9,,,6Dm-DMO1 090a

Upon the use of 2.20), we obtain the potential

I C2) V(O = 4e-2l(tr

I g2 (1 112) 2 _ g2 0)2 = e-2l 2 7tr (01T, (2.26)

8 Observe that since T are anti-hermitian, the second term is positive definite by itself, as is the first term. Rom the above potential, it is obvious that the absolute minimum is at O = . Thus -the potential has a unique minimum. There axe no moduli. Note that if we could set 1 = there could be other nontrivial configurations which could break E7 spontaneously. However in this -particular model g, has to be different from zero for the aa-lomaly cancellation. The nonvanishing of g, is also the basic reason why the manifold R4 X S2 iS.,L solution. Essentially, at the minimum 2 1 of the hyperscalars the exponential potential for the dilaton is given by 2gi e- 2 0- For a constant dilaton this acts like a 6-dimensional cosmological constant. When a U(l) gauge field assumes a magnetic monopole configuration on S2 we obtain the solution R4 X S2.

.By examining the susy transformation rules it becomes clear that there can be no product space

Solution of the form M4 x K2 with a nonzero vev of any of the non-Abelian gauge fields preserving any amounts of supersymmetries.

The fact that the minimum of the hyperscalar potential is at 0 implies that the E7 symmetry can not be broken spontaneously by a vev of the hyperscalaxs at the tree level. The only possibility of a tree level breaking of E7 (as well as E6) is to give a vev to the components of vector potential of these groups tangent to the internal manifold S2. If the monopole sits in

.E6 x E7 factor, the configuration is generally unstable, unless the monopole chaxge is chosen to be the least possible value 16]. Since such configurations also break all the D 6 supersymmetries it follows that at the tree level E6 x E7 and susy break at the same scale.

The mass of the fluctuations of the scalar fields around the minimum at - 0 will have two contributions for their masses. The first is the mass term coming from the potential in 2.26) and the second is the KK mass originating from the fact that the scalaxs are charged with respect to the U(1) x E7 gauge fields. Therefore a magnetic monopole background sitting in this group will generate a nonzero mass for all the scalars, unless the magnetic charge of the two groups cancel out mutually. If the effective magnetic coupling of a scalar field on S i n, then the mass squaxed of the lightest Kaluza Klein mode in the expansion of ok will be proportional to jnj/a2 where a is the radius of S2.

On the other hand since the fermions will couple to the magnetic monopole embedded in E7, there will be many chiral fermions; in the low energy spectrum in R4, exactly in the same manner as in [3] where the monopole in a U(1) subgroup of E6 gave rise to two families of 16 of SO(10) in R . Several models of this type in which the Higgs scalars may originate from the extra components of the gauge field have been studied in detail in [17]

Having established that the minimum of the hyperscalar potential is at 0 the proof of the uniqueness of the R' X S2 solution should follow along the same lines as given in 2] for the

Salam-Sezgin model.

9 3 The Ansatz, the F and H Field Equations and Supersymmetry

In this section we present the dyonic string ansatz and determine the equations that follow from the requirement of F and H field equations, and supersymmetry. Once these equations are satisfied, we show that the Einstein and dilaton field equations are automatically satisfied as well. We then proceed to solve all the required equations in section 4 where we present our dyonic string solution.

The Ansatz

Now we turn to the dyonic string ansatz. But before stating our ansatz we will briefly summarize all the known brane solutions in our model. In 2 it was shown that the most general solution of our model compatible with the Poincar6 symmetry in R 4 is a 3-brane with warped metric. The brane is a J-function singularity which can also be interpreted as a deficit angle in the 2-dimensional transverse space. This solution breaks all the supersymmetries. It reduces to 12 supersymmetric solution when the deficit angle vanishes. In 141 solutions of the tpe AdS3 x SI as well dyonic string solution have been studied. It has also been shown that the AdS3 x SI solution goes over to the maximally symmetric R4 X S2 configuration.

Our solution will be a generalization of the dyonic string of 14], in which in addition to the

U(1)R gauge field a U(1) component in E7 will also be nonzero, in a nontrivial way. We thus start from the following ansatz:

ds2 = c 2dx1A dx A + a 2 0,2 + 0,2) + b2 0,2 + h 2 dr2 6 1 2 3

H = PorjA6r2AU3+ud 2xAdr,

F3 = k a, A r2,

F7 = v or, A U2 + VU3 A dr , (3.1)

where d2x = dx" A dx'e,,,, k is a constant, a, b, c, h, u, v, P and o are functions of r, and the ai are left-invariant 1-forms on the 3-sphere, satisfying the exterior algebra

du = - 21 Eijk aj A Uk (3.2)

They can be represented, in terms of Euler angles (0, o,,O) by

U1 + iOr2 = e'O dO + i sinOdW), Or3 = do + cos 0 dW . (3.3)

The function h which may be removed by a coordinate transformation, dr = h(r) dr, will be chosen later to simplify the solution. Locally, we can choose the potential for F3 to be given by

1 0 A 3 = -kU3, and for P by A 7 = -V U3 - It is also useful o record

*F 3 = khbc2 d2X -2 93 A dr A

via 2 vhb *F 7 = C2 orl A Or2 + - - r Ad 2X ( hb a2 03Ad

*H uba2 Phc2 2 C2 01i A U2 A 0'3 d x A dr (3-4) ba2

'The F and H Field Equations

'The H-field equation and the Bianchi identity dH = 21 tr F 2 axe solved, respectively, by

QOhc2 p = p _ V2 u ba-2 e 0 2 (3.5)

Where Po and -QO axe integration constants. The F 3 and F7 field equations axe solved by

2 1 vOhb b POW V __ e (3.6) a21,2

'where vo is an integration constant.

.Killing Spinor Conditions

-.714ext, we examine the consequences of supersymmetry. Following 14], we impose the following conditions on the which break supersymmetry by a factor of four:

12 F EA = (T')A'EB I'1234 EA = EA (3.7)

,Thus, the conditions JA3 = 0 and 8A 7 = 0 give 1 2 keep V hb a 2gl _V = J2 (3.8)

The condition SOa = 0 is trivially satisfied, while JX = 0 is solved by

I IV U Ph' W - el (T2 + J) (3.9)

There remains the supersymmetry transformations of the gravitini. To this end, it is useful to note that in the orthonormal frame defined by

e0 = cdt , e 1 =c dx e1 = aa, e2 = a Or2 e3 = bU3 e4 = hdr the non-vanishing components of the spin connection take the form

b 1 b 2 ( b 3 W23 = - e , W31 = - e W12 = 2 e 2a2 2a2 2a b

aI 1 a 1 2 b 3 M el A W14 = ah e , W24 = a h e , W34 = bh e , W 4 = ch e 3-10)

11 Using these results, and taking E(r), it follows from 60, = 0, 6Oi = 1 2 and 3 = , respectively, that C, = 1w ( u - Ph) c :ie2 2 J (3.11) al = I I U Ph) bh a Te2 H + a 2 - ja2 (3.12)

b' = IO V u + Ph) bh (kg, - 1h (3.13) b T 2 aj J b

Finally, 604 0 gives 1 U Ph E 1 e2w - (3.14)

Comparing with 3-11), we learn that

C(r) = 1/2 EO . (3-15)

The Einstein and Dilaton Field Equations

Let us begin by writing the Einstein and dilaton equations given in 2.23 as

RMN = TMN , DW=J. (3-16)

Substitution of the ansatz into these equations yields rather complicated field equations which have provided in the Appendix. Instead of solving these complicated second order field equations, it is much easier to show that they axe automatically satisfied once the Killing spinor conditions, and the F and H-field equations/Bianchi identities are satisfied. To see this, let us first introduce the notation J'OM = bme 6X = Ae (3-17)

It is then straightforward to show that

rN[_bM, bNjC (RmN - TMN) r N_ + XM

rm[f5m, A],- = (D o - J) - + Y , (3.18) where XM and Y are expressions which vanish upon the use of the F and H field equations/Bianchi identities. Therefore, the dilaton equation is evidently satisfied, and so is the

Einstein equation, once we note that RMN is diagonal for our ansatz, as shown in the Ap- pendix.

12 4 The Dyonic String Solution

As in 14], we make the gauge choice

h 2a2bC2 r3

Then, defining the combinations W± = o ± 41nc , (4.2)

we find from 3.9) and 3.11), with the help of 3.5),(3.6.) and 3.8) that 4QO 1 - 3 e-2w-

4PO IV+ _ 02 1 e2 2 + -3 e-- VO (4.3) NF2 Ro

These have solutions

eV OH, coth (OH2) c 4 = Hi tanh (OH2) (4.4)

where

Hi = (o + Q0r2 H2 = AO + ;2Po (4.5) Here, we have introduced the integration constants Q0 nd Po. Next, from 3.13), making use

of (3.5), 3.6), and 3.8) we find PO = k(l - kgi) (4.6) 2gl Note that 3.5), 3.6), and 3.8) also yield the results I voe 2v' PO P cosh 2 (OH2) (4.7)

The remaining quantities in the ansatz, namely, the functions (a, b, u) can now be evaluated in terms of (V, c) via algebraic equations 3.8), 3.6), 3.5) and 4.7). The result for the ansatz can now be summarized as follows:

2 dxPdx, k (,2 + 2) + PO .2 p0#2 P dr2 ds6 FOH1 coth-OHj [ HI + W 1 2 cosh2(#H2) -3 +g2 2(,6 - -6 1 sinh H2) eV = OH, coth (OH2)

H = PO orl A Or2 A Or3 -d2x A dH1_1 cosh2 (OH2)

F 3 = k or, A C72 , 7 = - 16 V,Mo (tanh (OH2) a, A Or2 + cosh2(OH2) U3 A dH2) (4.8)

13 with 4.6) holding, and (HI, H2) and are defined in 4.5) and 4.3), respectively, and (PO, k, vo) axe constants. Furthermore, from 3.6) 38),(4.6) and 4.7), we learn that 1 Po k < (4.9) 91

In the Emit of 8 0, the U (1) C E7 gauge field vanishes and the non-vanishing fields become

p2 21 2 1 k 2 2) p 0 dr ds 6 dxPdx,, + or 1 + or 2 0 Cr23 + 2(H2)2 r6 H2 [Hi 91 91

eV HI H2

P0a1A0'2AU3-d 2 x A dH-1

F3 ka A U2 , (4.10) with 4.6) holding. This is the solution obtained in 14].

Turning to our solution 4.8), in order to study its global structure, following 14] we change to a new radial coordinate p related to r as p p2 (4.11) r2 P4

Our solution then is given by 4.8) with

HI = (o _ QoPo) + QoPo H2 = Po2 (4.12) PO P4 P4

We now observe that both our metric 4.8) as well as its 0 limit given in 4.10) take the same form at spatial infinity reached by taking the p -+ oc limit, and the resulting metric is

2 - dp2 + gIP0 P2 (U2+U2+291PO U2 + 2g' dxtdx, (4.13) ds6 - 2PO2 8k 1 2 k 3 kQo 91 ) I where Qo = (QoPo - QoT0) /Po. This metric indeed describes a cone over the product of

Minkowski2 X the squashed 3-sphere 14]. In this limit F7 vanishes, F3 and H are finite but the

_+ Q 4. dilaton diverges as eV Po2 P Our metric 4.8) has a horizon at p = 0, just as its 0 -+ 0 limit given in 4.10) does. In this limit we obtain

-20Po2 2p2po 20p2 2 2 kQoPo 2 2 4 2 2) 4k dp2 ds6 P dxudx,, + P2 U1 + or2 + 2g1P0 e P or 3 + g2 e__i4' 1 40-po 2gl k 1 (4.14)

Furthermore, while H vanishes and F3,F7 become constants, the dilaton diverges as eV

0Q0p01p4 in this limit. Interestingly, taking the vo -+ 0 limit of this near horizon metric (4.14)

14 does not yield the same result as first taking such a limit in the full metric 4.8) and then going to the horizon at p = 0. In the latter case, as shown i 14], one obtains a direct product of AdS3 with squashed 3-sphere.

5 Discussion

In this paper we have given the precise form of the potential for the scalars in the hypermatter multiplet of the only known anomaly free gauged (1, 0) supergravity in D = 6 in the absence of

linear multiplets. The model has the gauge group of E13 x E7 X MR. The hyperscalars axe charged with respect to MR and transform in the 912 dimensional pseudo real representation of E7. They are singlets of E6. We showed that the potential has a unique minimum at = . Despite the fact that there is no obvious mass term in the D=6 action for the scalars their

Kaluza Klein tower, on a background of R 4 x K2 will The all massive due to their U(1 x E7 charges. The hypermatter fermions on the other hand will give rise to plenty of chiral fermions in D = 4 Thus opening the road for a detailed phenomenological study of our model.

One interesting direction in further study of our model would be cosmological investigation

along the lines of [18, 19, 20]. The quantum loop of the massive scalars in this model axe the natural candidates to stabilize the radius of the compact space in a cosmological context at finite temperature. It has been shown long time ago that the effect of such loops can generate a constant radius for the internal space while the scale factor of our 3-dimensional universe expands according to the standard Friedman Robertson 'Walker law 21]. It is a very interesting question to seek for an accelerating universe solution. Such solution should exist according to the criteria given in 22].

'In this paper we also constructed a dyonic string solution in the same model. Our solution leaves 14 of the original supersymmetries, i.e. one complex supersymmetry in 1 + 1 dimensions, unbroken. Furthermore a U(1) component of the E7 gauge field needs to be nonzero. In fact it assumes a rather complicated form. The solution approaches a cone as r -+ 00 over a squashed

'53 X Minkowski2, while at r = 0 it has a horizon.

Another important question is to complete the search iitiated in 7), for a higher dimensional origin of this gauged supergravity model in D = 6.

Acknowledgments

IvVe are very grateful to John Strathdee for his collaboration at the early stages of this work. We thank Gary Gibbons, Rahmi Giiven, Jim Liu, Hong Lu and Chris Pope for useful discussions. E.S. would like to thank the Abdus Salarn International Center for Theoretical Physics for hospitality. The work of E.S. is supported in part by NSF grant PHY-0314712.

15 Appendix The Anomaly Polynomial There are few misprints in the anomaly polynomial formulae of 3] which we wish to correct here. We begin by listing the individual contributions (here, F6, F7, F, denote the E6, E7, U(1)R field strengths): 192trR 2 + 1 [245 tr R4 _ 5 x 43 (tr R2)2] F1 - F, (5.1) 24 96 5760 4

1 4 + 1 2 + 912 4 + 2)2] 2 trR [tr R (trR (5.2) -2P(V)R = 24 Tr912 F 96 Tr912 F7 5760 4

14 + 1 2trR2 + 1 [tr R4 + 5 (tr R2)2] (5.3) -P(XR = 24 F, 96 F, 5760 4

P(AL = 1T-78 F64 + 6 Tr7,8F2F2 + 78F4) 24

+ 'Ih33 F 4 + 6 Tr133 F72F2 + 133F4) 24 7

2 [a78 F62 + rR133 F72 + (78 + 133 + 1)F2 tr R 96

(78 + 133 + 1) 4 + 5 2)2 tr R 4 (trR (5.4) 5760 1 1 -

Using the relations between traces in the adjoint and fundamental representations provided in [31 and adding all the contributions, we find that P = fl'Op) + P('OR) + P(XR) + P(I\R) is given by

1 2)2 + 1 2 2- 2 'Ir56F72 + 2 trR2F2 P 16 (trR 24 trR Tr27F6 8trR

* 1'IY27 (F62)2 + tr27F62FJ2 -3 M.6 F72)2 48 64 3 F2 2 4 * rR5 6 7 F F, (5.5) 4

The signs of the (trR2)2 terms in 5.2), (5.3), 5.4) and a factor of two in the coefficient of trR 2a56F72 in (5.5) have been corrected.

It is possible to factorize this expression and write it as

1 2 2 + 1 2 + 1 2) ( trR 2- 36F2 - tr27F62 + 3 2) (5.6) P 16 ( t7-R + 4F, 3 tr27F6 2 tr56F7 2 tr56F7

16 The Einstein and Dilaton Field Equations

Writing the Einstein's equation for the model as Rr = S, where, we recall that r, s 0, 1, 1 2 3 4 label the tangent space frame defined in 3 the non-vanishing components of Rr, evaluated for the ansatz 3.1) are

R11 = C,2 2a'c' Yc' I l(c' 77/LV c2 a h2 + c h2 +ch h),

2a'c' a' bl a/2 1 (a) P RI, = R22 ach2 abh2 a2 h2 ah h 0 + a2

2b'c' 2a' Y 1 Y b2 R33 = - -ch2 a b h2 - bh h 2 4

R44 = - 2 (a), - 1 (b) _ 2 cl, (5.7) ah Th h Th c h h')

While the non-vanishing components of S, take the form

1 I V2 Vf2 1 U:2 p2 1g2 e2w - b2 eV - e - C V S14V J a4 p- )2 4 T2 j + 4V 2 2 77AV

2 2 t2 2 2 1 3k 3v V U P 2 SII S22 e2V + e2w + . e + 1 _a 4 (ga-4 Th YJY 4 T2-C4 + 4 b2 29 1 e 2 I 2 2 p2 1 k V 30 1 + I U . V 2 w S33 g-a4 e 2 4 e e 9 e a h_2b2 4 )2,C.4 + 4_b2 2 k2 1 V2 30 I U2 P2 2 S44 e2w e2w eV + 19 e 2w (5.8) 9_0 0 9-h-22b ) 4 T2 -C.4 + -a4b2 2

Finally, writing the dilaton field equation as 0 V = J, he evaluation of J for our the ansatz (3. 1) yields

P V2 Vi2 1 P2 U2 1 e2w + + b2 e W + eV - 2 2 e 2w (5.9) W a4 T2-- ) 2 4V h2 C4 1

17 References

[1] A. Salam and E. Sezgin, Phys. Lett. 147, 47 1984).

[2] C. W. Gibbons, R. Giiven and C. N. Pope, axXiv:hep-th/0307238.

[3] S. Randjbar-Daemi A Salam, E. Sezgin and J. Strathdee, Phys. Lett. 151, 351 1985).

[4] H. Nishino and E. Sezgin, Nucl. Phys. B 278, 353 1986).

[5] H. Nishino and E. Sezgin, Nucl. Phys. B 505, 497 1997), arXiv:hep-th/9703075.

[6] M. Berkooz, R. G Leigh, J. Polchinski, J. H Shwaxz, N. Seiberg and E. Witten, Nucl. Phys. B 475, 115 1996), arXiv:hep-th/9605184.

[7] M. Cvetic, G. W. Gibbons and C. N. Pope, arXiv:hep-th/0308026.

(8] T. Gherghetta and A. Kehagias, Phys. Rev. D 68, 065019 2003), arXiv:hel)-th/0212060.

[9] Y. Aghababaie, C. P. Burgess, S. L. Parameswaran and F. Quevedo, axXiv:hep-th/0304256.

[10] S. Randibar-Daemi, A. Salam and J. Strathdee, Nucl. Phys. 24, 491 1983).

[1 1] S. M. Carroll and M. M. Guica, arXiv:hep-th/0302067.

[121 I. Navarro, JCAP 0309, 004 2003), arXiv:hep-th/0302129. Class. Quant. Grav. 20, 3603 (2003), arXiv:hep-th/0305014.

[13] G. Dvali G Gabadadze and M. Shifman, Phys. Rev. D 67, 044020 2003), arXiv:hep- th/0202174.

(14] R. Gilven, J. T. Liu, C. N. Pope and E. Sezgin, arXiv:hep-th/0306201.

[15] R. Percacci and E. Sezgin, arXiv:hep-th/9810183.

[16] S. Randibax-Daemi, A. Salam and J. Strathdee, Phys. Lett. 124, 345 1983) [Erratum- ibid. B 144, 455 1984)].

[17] G. R. Dvali, S. Randjbar-Daemi and R. Tabbash, Phys. Rev. D 65 2002) 064021, axXiv:hep- ph/0102307.

[18] J. M. Cline, J. Descheneau, M. Giovannini and J. Vinet, JHEP 0306, 048 2003), arXiv:hep- th/0304147.

[19] T. Bringmann and M. Eriksson, JCAP 0310, 006 2003), axXiv:astro-ph/0308498.

[20] T. Bringmann M Eriksson and M. Gustafsson, Phys. Rev. D 68, 063516 2003), arXiv:astro-ph/0303497.

[21] S. Randjbax-Daemi, A. Salam and J. Strathdee, Phys. Lett. 135, 388 1984).

[22] P. K. Townsend, arXiv:hel>-th/0308149.